I did not understand the reasoning behind why a=b is not valid.
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For the equation to be a circle the value of the coefficients of x^2 and y^2 must be the same (if both are different then it will be an ellipse). Hence a=b is not valid because then ax^2+bx^2+2cx+2dy-2k=0 doesn't represent a circle. It will only represent a circle when a=-b (not +b).
I hope this helps!